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Abstract

In these notes we are going to present some technique which is a multidimensional analogue of some methods which are nowadays standard in scattering theory on the real line for the Schrödinger operator. These methods are based on the construction of operators intertwining the Schrödinger operator with the ‘free operator’ obtained when the potential term is removed. We refer to the monograph [5] by V. A. Marchenko and to the paper [6] for a detailed presentation of this technique.

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References

  1. L. D. Faddeev. Inverse problems of quantum scattering theory. ii. J. Soviet Math., 5: 334–396, 1976.

    Article  MATH  Google Scholar 

  2. L.D. Faddeev. Uniqueness of the solution of the inverse scattering problem. Vestnik Leningrad. Univ. Mat. Mekh. Astronom., 7: 126–130, 1956.

    MathSciNet  Google Scholar 

  3. L. Hörmander. The analysis of linear partial differential operators. Springer-Verlag, Berlin, Göttingen, Heidelberg, New York, Tokyo, 1983–1985.

    Google Scholar 

  4. R. Lagergren. The inverse back-scattering problem (preliminary title). Technical report, Department of mathematics, Växjö University, 2000.

    Google Scholar 

  5. V.A. Marchenko. Sturm-Liouville operators and applications, volume 22 of Operator theory: Advances and applications. Birkhüser Verlag, Basel-BostonStuttgart, 1986.

    Google Scholar 

  6. A. Melin. Operator methods for inverse scattering on the real line. Comm. Partial Differential Equations, 10: 677–766, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Melin. Intertwining methods in the theory of inverse scattering. Internat. J. Quantum Chem., 31: 739–746, 1987.

    Article  Google Scholar 

  8. A. Melin. On the general inversion problem. In Brändas, E. and Slander, N., editors, Resonances, volume 325 of Lecture Notes in Physics, pages 47–55. Springer-Verlag, 1989.

    Google Scholar 

  9. A. Melin. On the use of intertwining operators in inverse scattering. In Holden, H. and Jensen, A., editors, Schrödinger operators, volume 345 of Lecture Notes in Physics, pages 383–400. Springer-Verlag, 1989.

    Google Scholar 

  10. A. Melin. The inverse scattering problem for a quantum mechanical two-body system. In Gyllenberg, M. and Persson, L.E., editors, Analysis, algebra and computers in mathematical research (proc of 21: s1 Nordic congress of mathematicians), Lecture notes in pure and applied mathematics, pages 247–262. Marcel Dekker, 1994.

    Google Scholar 

  11. A. Melin. The Faddeev approach to inverse scattering. In Hörmander, L. and Melin, A., editors, Partial differential equations and mathematical physics, Progress in nonlinear differential equations and thir applications, pages 226–245. Birkhäuser, 1996.

    Google Scholar 

  12. A. Melin. Back-scattering and nonlinear radon transform. Sémin. Equ. Dériv. Partielles, Ecole Polytechnique, pages X IV, 1–14, 1999.

    Google Scholar 

  13. A. Melin. Intertwining methods in direct and inverse scattering. 2000. monograph in preparation.

    Google Scholar 

  14. R.G. Newton. Inverse Schrödinger scattering in three dimensions. Texts and monographs in Physics. Springer-Verlag, 1989.

    Google Scholar 

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Holst, A., Melin, A. (2004). Intertwining Operators in Inverse Scattering. In: Bingham, K., Kurylev, Y.V., Somersalo, E. (eds) New Analytic and Geometric Methods in Inverse Problems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-08966-8_2

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  • DOI: https://doi.org/10.1007/978-3-662-08966-8_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-07379-3

  • Online ISBN: 978-3-662-08966-8

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